Gaps in Mesh After Applying Modifiers
The misleading representation of wires does not mean that your mesh has holes.

The inconsistent display of wires for faster viewport display can be "corrected" with the Draw All Edges option.

As you noted correctly, after the boolean modifier, the wire display is not consistent with the solid representation.


However, adding wire display in the Object's Properties Panel, shows the same wires in solid view.


To show all wires, enable Draw All Edges.




There are, however, many other tools, which are useful for cleaning objects. For example:  


the Mesh Analysis Section of the Properties Panel in the 3D View

the "Select Non Manifold Geometry" option in the 3D View &#9096; Ctrl&#8679; Shift&#9095; AltM
the 3D print toolbox addon